Periodically Depleted Honeycomb Lattice

• Periodically depleted honeycomb lattices are networks where regular site removal enlarges the unit cell and alters connectivity while preserving long-range translational order.
• These structures enable control over electronic band features, such as flat bands and Dirac-like crossings, through engineered hopping modulations and selective defects.
• They offer practical insights into tuning topological phases and modeling correlated states in systems like strained graphene and frustrated magnetic materials.

Searching arXiv for recent and foundational papers on periodically depleted honeycomb lattices and closely related depleted-lattice constructions. A periodically depleted honeycomb lattice is a lattice obtained from the honeycomb network by removing sites, bonds, or symmetry-equivalent degrees of freedom in a regular superlattice pattern, so that translational order is retained while the unit cell is enlarged and the connectivity is altered. In the literature surveyed here, the term encompasses several distinct but related constructions: a $1/6$-depleted honeycomb lattice known as the Bishamon-kikko (BK) lattice in topological band theory (Ikegami et al., 2024); selective random removal of only the sites that would be absent in the BK pattern, which interpolates between the pristine honeycomb and the periodically depleted limit (Ikegami et al., 18 Nov 2025); depletion-induced anisotropic honeycomb analogues relevant to strained graphene and depleted square-lattice mappings (Guo et al., 2016); and emergent honeycomb backbones generated by periodic reconstruction from other parent lattices (Flint et al., 2013). In magnetic settings, depletion can also mean site dilution by non-magnetic dopants on the honeycomb lattice, which replaces clean zigzag order by spin-glass freezing with short-ranged zigzag correlations (Andrade et al., 2013). Across these realizations, periodic depletion functions as a controlled modification of lattice connectivity that reorganizes spectra, topology, and collective ordering phenomena.

1. Geometric construction and lattice-theoretic meaning

The most explicit periodically depleted honeycomb construction in the cited literature is the BK lattice, obtained by periodically removing $1/6$ of the sites from the honeycomb lattice (Ikegami et al., 2024). The remaining sites form a lattice with threefold rotational symmetry $C_3$ and a larger unit cell than the pristine honeycomb lattice. Because the deletion is periodic rather than random, Bloch band theory remains applicable, and the resulting lattice supports both dispersive bands and states localized on defect-related sites (Ikegami et al., 2024).

A related construction appears in the Bishamon-kikko–honeycomb (BKH) model, where the depleted BK limit and the ordinary honeycomb limit are connected continuously by modulating hoppings involving the defect sites through a parameter $\alpha$ (Ikegami et al., 2024). In that formulation, $\alpha=1$ reproduces the full honeycomb Haldane model, while $\alpha=0$ disconnects the defect sites and yields the BK lattice with isolated defects (Ikegami et al., 2024). This provides a periodic rather than stochastic route between the two geometries.

A second notion of periodic depletion arises through mappings from other lattices. The 1/3-depleted square lattice studied in the Hubbard and Heisenberg contexts is described as equivalent to a strained honeycomb lattice with two inequivalent bond types (Guo et al., 2016). Here the depletion is performed on a square lattice by removing every third diagonal stripe of sites, leaving a bipartite network that can be viewed as a honeycomb-like lattice with anisotropic hoppings or exchanges (Guo et al., 2016). The resulting geometry is not a depleted honeycomb in the strict site-removal-from-honeycomb sense, but it is a periodically depleted honeycomb analogue in the sense used for strained graphene.

A third form is the emergent honeycomb lattice proposed for LiZn$_2$Mo$_3$O$_8$, where a structural distortion of a triangular lattice effectively separates the original spins into two sublattices forming a honeycomb network plus a third set of orphan spins (Flint et al., 2013). The paper explicitly characterizes the $x=1$ limit as a depleted honeycomb lattice plus decoupled central spins, with the reconstruction tied to a regular pattern that triples the unit cell while preserving trigonal symmetry (Flint et al., 2013).

These constructions share a common structural feature: periodic depletion changes coordination, unit-cell multiplicity, and sublattice balance without destroying long-range spatial order. This suggests that the periodically depleted honeycomb lattice is best understood as a family of superstructured honeycomb-derived networks rather than a single unique graph.

2. Canonical Hamiltonians on periodically depleted honeycomb lattices

The main electronic model used on the BK lattice and its interpolating generalizations is the Haldane model for spinless fermions (Ikegami et al., 2024). On the BK lattice, the Hamiltonian is

$1/6$0

with staggered onsite potential

$1/6$1

and complex second-neighbor hoppings carrying phases $1/6$2 depending on clockwise or counterclockwise orientation (Ikegami et al., 2024). The paper sets $1/6$3 as the energy unit (Ikegami et al., 2024).

For the interpolating BKH model, hoppings involving defect sites are scaled by $1/6$4, with $1/6$5 if a hopping process includes a defect site and $1/6$6 otherwise (Ikegami et al., 2024). The corresponding Hamiltonian is

$1/6$7

This same hopping-modulation idea is later recast as an effective description of selective random defects, with the paper suggesting the approximate correspondence $1/6$8 between the periodic modulation parameter and the defect ratio $1/6$9 (Ikegami et al., 18 Nov 2025).

On the magnetic side, the depleted honeycomb lattice enters through frustrated spin models for iridates. One case is the Heisenberg-Kitaev model,

$C_3$0

where $C_3$1 labels bond type (Andrade et al., 2013). Another is the Heisenberg $C_3$2-$C_3$3-$C_3$4 model,

$C_3$5

used to model zigzag order in $C_3$6 and related compounds (Andrade et al., 2013). In that work, depletion is introduced by randomly removing a fraction $C_3$7 of magnetic sites, producing

$C_3$8

remaining spins on an $C_3$9 layered system (Andrade et al., 2013).

The strained-graphene analogue on the $\alpha$0-depleted square lattice employs both Heisenberg and Hubbard Hamiltonians with two inequivalent couplings (Guo et al., 2016). In the localized-spin limit,

$\alpha$1

while the half-filled itinerant model is

$\alpha$2

with the strong-coupling relation $\alpha$3 linking hopping anisotropy to exchange anisotropy (Guo et al., 2016).

3. Band structure, flat bands, and topological phases

Periodic depletion modifies the band count per unit cell and thereby enriches the topological band structure. On the BK lattice at $\alpha$4, the Haldane model exhibits a flat band at energy $\alpha$5 for $\alpha$6, $\alpha$7, originating from localized states on the defect/yellow sites via destructive interference (Ikegami et al., 2024). In the same regime, the upper dispersive band touches the flat band quadratically at $\alpha$8, while the dispersive bands show Dirac-like linear crossings at the $\alpha$9 point (Ikegami et al., 2024). When $\alpha=1$0 with complex phase $\alpha=1$1, the flat band becomes dispersive, the Dirac nodes at $\alpha=1$2 are gapped, and Chern insulating phases appear (Ikegami et al., 2024).

The BK lattice supports a richer topological phase diagram than the original honeycomb Haldane model because its enlarged unit cell produces commensurate fillings $\alpha=1$3, $\alpha=1$4, $\alpha=1$5, and $\alpha=1$6 rather than the honeycomb’s simpler half-filled structure (Ikegami et al., 2024). The phase diagrams are not symmetric under $\alpha=1$7, reflecting the inequivalent sublattice structure created by periodic depletion (Ikegami et al., 2024). A notable consequence is the appearance of insulating phases with

$\alpha=1$8

which do not occur in the standard honeycomb Haldane model (Ikegami et al., 2024). The same study also identifies metallic regions with noninteger $\alpha=1$9, indicating substantial Berry-curvature structure even in the absence of a full bulk gap (Ikegami et al., 2024).

The topological invariant is the occupied-band Chern number,

$\alpha=0$0

computed numerically by the Fukui–Hatsugai–Suzuki method (Ikegami et al., 2024).

The interpolating BKH model shows that depletion is not merely a singular geometric endpoint but a continuous tuning parameter for topological structure. At fixed $\alpha=0$1, varying $\alpha=0$2 changes the competition between localized defect-band physics and ordinary honeycomb Haldane physics (Ikegami et al., 2024). At $\alpha=0$3, a flat band at $\alpha=0$4 appears from isolated defect sites, which alters the topological accounting at fillings $\alpha=0$5, $\alpha=0$6, and $\alpha=0$7 (Ikegami et al., 2024). As $\alpha=0$8 increases from 0 to 1, the $\alpha=0$9 insulating regions generally shrink and metallic regions expand through metal-insulator transitions (Ikegami et al., 2024).

A mathematically distinct periodically depleted honeycomb setting is the honeycomb lattice with periodically arranged impenetrable obstacles, treated through Dirichlet and Neumann eigenvalue problems (Li et al., 2022). There, periodic depletion means puncturing each cell by a circular obstacle $_2$0, yielding a punctured cell $_2$1 (Li et al., 2022). The resulting Bloch spectrum exhibits Dirac points at $_2$2 and at higher band crossings, with conical splitting

$_2$3

and slopes reciprocal to the eigenvalue $_2$4 (Li et al., 2022). This establishes that periodic depletion need not mean literal site deletion in a tight-binding graph; it may also denote periodic removal of accessible spatial regions in continuum honeycomb media.

4. Disorder, selective depletion, and interpolation between periodic limits

A central recent development is the distinction between generic disorder and selective random defects placed only on the subset of sites that would be removed in the periodic BK pattern (Ikegami et al., 18 Nov 2025). This construction defines a defect ratio

$_2$5

with $_2$6 the pristine honeycomb lattice and $_2$7 the fully depleted BK lattice (Ikegami et al., 18 Nov 2025). The intermediate regime $_2$8 does not produce arbitrary vacancy disorder; instead, it generates a constrained random interpolation between two clean periodic structures (Ikegami et al., 18 Nov 2025).

The electronic Hamiltonian remains the Haldane model,

$_2$9

with $_3$0 on the A sublattice and $_3$1 on the B sublattice, and $_3$2 as the energy unit (Ikegami et al., 18 Nov 2025). Because translational symmetry is absent for $_3$3, the topology is characterized in real space using the local Chern marker, crosshair marker, and Bott index (Ikegami et al., 18 Nov 2025).

The local Chern marker is

$_3$4

with $_3$5 and $_3$6 (Ikegami et al., 18 Nov 2025). The crosshair marker is

$_3$7

and the Bott index is

$_3$8

with projected position operators

$_3$9

(Ikegami et al., 18 Nov 2025).

Two distinct regimes are identified. For

$_8$0

the density of states and energy spectrum evolve continuously from honeycomb to BK, and the local Chern marker, crosshair marker, and Bott index all remain

$_8$1

when the chemical potential lies in the relevant gap (Ikegami et al., 18 Nov 2025). In this regime, the two periodic limits are topologically smoothly connected.

For

$_8$2

the interpolation instead produces a bulk-gap closing near $_8$3, around

$_8$4

followed by reopening; under open boundary conditions, in-gap edge states are present for $_8$5 and absent for $_8$6, while all three real-space invariants jump sharply at the same $_8$7 (Ikegami et al., 18 Nov 2025). The work therefore identifies a disorder-induced topological transition caused by constrained depletion rather than generic randomness.

The same paper proposes an effective periodic model in which selective random defects act as hopping-amplitude modulation: $_8$8 with

$_8$9

For $x=1$0, $x=1$1, $x=1$2, the effective model has a Dirac crossing at the $x=1$3 point when

$x=1$4

matching the disordered-lattice transition at approximately $x=1$5 and supporting the approximate mapping $x=1$6 (Ikegami et al., 18 Nov 2025). This suggests that certain forms of selective random depletion can be interpreted as renormalized periodic connectivity rather than mere disorder broadening.

5. Magnetism, correlations, and order-disorder phenomena

In frustrated honeycomb-lattice iridates, depletion acts as a probe of exchange range and magnetic frustration. In the study of diluted Heisenberg-Kitaev and $x=1$7-$x=1$8-$x=1$9 models for $1/6$00 ($1/6$01), non-magnetic doping is implemented by replacing a fraction $1/6$02 of magnetic $1/6$03 ions with inert dopants, removing spin sites from the honeycomb network (Andrade et al., 2013). In the clean systems, both models can stabilize zigzag order (Andrade et al., 2013). With dilution, however, the clean zigzag state is generically replaced by a spin-glass state with short-ranged zigzag correlations (Andrade et al., 2013).

The work diagnoses this regime through the magnetic correlation length near the zigzag ordering wavevectors and the spin-glass correlation length from the spin-glass susceptibility (Andrade et al., 2013). The overlap order parameter between replicas is

$1/6$04

and

$1/6$05

The freezing temperature $1/6$06 is extracted from finite-size crossings of $1/6$07 using

$1/6$08

with $1/6$09 in the reported analysis (Andrade et al., 2013).

The key physical result is the contrast between short-range and longer-range exchange near the site-percolation threshold of the honeycomb lattice,

$1/6$10

For the effectively short-ranged Heisenberg-Kitaev model, $1/6$11 falls rapidly as $1/6$12 approaches $1/6$13 and becomes extremely small just above it (Andrade et al., 2013). By contrast, in the $1/6$14-$1/6$15-$1/6$16 model the glass temperature remains appreciable well beyond $1/6$17, extrapolating toward much larger dopings around $1/6$18 (Andrade et al., 2013). The proposed experimental implication is explicit: if the freezing temperature collapses near $1/6$19, the magnetism is predominantly short-ranged; if it persists beyond $1/6$20, substantial longer-range exchange is indicated (Andrade et al., 2013). In this sense, depletion is used diagnostically rather than merely destructively.

A different magnetic manifestation of periodic depletion appears in the 1/3-depleted square lattice, viewed as a strained honeycomb analogue (Guo et al., 2016). In the Heisenberg limit, stochastic series expansion quantum Monte Carlo finds a quantum phase transition from antiferromagnetic long-range order to a disordered phase as exchange anisotropy grows, with critical coupling

$1/6$21

whereas linear spin-wave theory yields

$1/6$22

substantially overestimating the stability of order (Guo et al., 2016). In the half-filled Hubbard model, the antiferromagnetic structure factor

$1/6$23

tracks the transition from semimetallic or metallic behavior toward a band-insulating regime, but the paper emphasizes that loss of antiferromagnetic order is not explained solely by the vanishing density of states; local singlet formation is essential (Guo et al., 2016).

The noninteracting band structure in that model,

$1/6$24

has a band gap $1/6$25 that vanishes for $1/6$26, becomes semimetallic with Dirac points at $1/6$27, and turns into a band insulator for $1/6$28 (Guo et al., 2016). This establishes a link between periodically depleted honeycomb analogues, Dirac physics, and interaction-driven order.

6. Emergent and continuum variants

The concept of periodic depletion extends beyond literal site removal on a honeycomb graph. In LiZn$1/6$29Mo$1/6$30O$1/6$31, the proposed low-temperature state is not a preexisting honeycomb lattice but a self-organized depletion of a triangular lattice into an emergent honeycomb network plus orphan spins (Flint et al., 2013). The effective Hamiltonian is

$1/6$32

with $1/6$33, $1/6$34, and $1/6$35 unchanged, where $1/6$36 yields the depleted honeycomb limit plus decoupled $1/6$37-sublattice spins (Flint et al., 2013). The proposal is motivated by the experimental observation that the low-temperature Curie constant satisfies

$1/6$38

indicating that only one-third of the moments remain free while the other two-thirds enter a correlated collective state (Flint et al., 2013).

The paper further argues that orphan spins stabilize the sublattice-pairing spin liquid (SPS) more strongly than competing Néel or valence-bond-solid states, with second-order energy shift

$1/6$39

compared with

$1/6$40

(Flint et al., 2013). A plausible implication is that periodic depletion can create not only altered band topology or weakened order, but also entirely new low-energy lattices that favor quantum-disordered phases.

A further generalization appears in the continuum model of honeycomb lattices with periodically arranged impenetrable obstacles (Li et al., 2022). The periodic obstacle array preserves honeycomb symmetry and Brillouin-zone structure while depleting real-space accessibility within each unit cell (Li et al., 2022). The existence of Dirac points for both Dirichlet and Neumann problems shows that periodic depletion in continuum media can retain or regenerate canonical honeycomb spectral features, provided the symmetry and resonant structure are preserved (Li et al., 2022).

7. Conceptual synthesis and recurring themes

The surveyed literature does not treat the periodically depleted honeycomb lattice as a single universal model. Instead, it presents a recurring design principle: start from the honeycomb geometry or a parent lattice that can be reorganized into a honeycomb backbone, then modify connectivity periodically or selectively so that the new structure remains ordered but acquires a larger unit cell, altered sublattice structure, and new low-energy modes.

Several recurrent consequences emerge.

Band multiplication and flat-band formation: The BK lattice has more bands per unit cell than the pristine honeycomb lattice and supports a flat band from localized defect-site states (Ikegami et al., 2024). This provides a direct route to higher Chern numbers and metallic states with nontrivial Berry curvature (Ikegami et al., 2024).

Interpolation as a control parameter: Both the BKH model and the selective-random-defect model show that depletion can be tuned continuously between honeycomb and periodically depleted limits, either by a hopping parameter $1/6$41 or by a defect ratio $1/6$42 (Ikegami et al., 2024, Ikegami et al., 18 Nov 2025). Depending on parameters, this interpolation can be topologically smooth or can force a gap closing and a phase transition near $1/6$43 (Ikegami et al., 18 Nov 2025).

Order suppression versus order diagnosis: In frustrated magnetic honeycomb models, depletion destroys long-range zigzag order and promotes glassiness (Andrade et al., 2013). Yet the way the freezing temperature behaves near the percolation threshold distinguishes short-range from longer-range exchange (Andrade et al., 2013). This suggests that depletion is simultaneously a perturbation and a spectroscopic probe of interaction range.

Analogue and emergent realizations: The $1/6$44-depleted square lattice provides a bipartite anisotropic honeycomb analogue relevant to strained graphene and nickelates (Guo et al., 2016), while LiZn$1/6$45Mo$1/6$46O$1/6$47 provides an example in which a depleted honeycomb network may emerge dynamically from a triangular lattice (Flint et al., 2013). A continuum obstacle array gives yet another realization in wave physics (Li et al., 2022).

Misconception to avoid: Periodic depletion is not synonymous with random vacancy disorder. In the BK and BKH constructions, depletion is symmetry-controlled and preserves Bloch periodicity (Ikegami et al., 2024). In the selective-defect interpolation, even randomness is constrained to a particular removable subset and can mimic hopping renormalization rather than generic disorder smearing (Ikegami et al., 18 Nov 2025). Conversely, in the iridate spin-glass study, depletion is genuinely random site dilution and leads to quenched disorder physics (Andrade et al., 2013). The distinction between periodic, selective, and generic depletion is therefore essential.

Taken together, these works establish the periodically depleted honeycomb lattice as a unifying motif across topological band theory, frustrated magnetism, correlated-electron models, and continuum spectral problems. Its significance lies not in a single canonical phase, but in the systematic way periodic removal or decoupling of honeycomb degrees of freedom reshapes connectivity and thereby reorganizes topology, criticality, and collective order.